Being motivated by some methods for construction of homothetically indecomposable polytopes, we obtain new methods for construction of families of integrally indecomposable polytopes. As a result, we find new infinite families of absolutely irreducible multivariate polynomials over any field F. Moreover, we provide different proofs of some of the main results of Gao. | Turk J Math 33 (2009) , 283 – 294. ¨ ITAK ˙ c TUB doi: Integral and homothetic indecomposability with applications to irreducibility of polynomials ¨ Fatih Koyuncu and Ferruh Ozbudak Abstract Being motivated by some methods for construction of homothetically indecomposable polytopes, we obtain new methods for construction of families of integrally indecomposable polytopes. As a result, we find new infinite families of absolutely irreducible multivariate polynomials over any field F . Moreover, we provide different proofs of some of the main results of Gao [2]. Key Words: Absolute irreducibility, polytopes, integral indecomposability. 1. Introduction Absolutely irreducible polynomials over a field are important and have applications in many areas of algebra and geometry such as algebraic geometry, number theory, coding theory, combinatorics, permutation polynomials. We have some well-known irreducibility criteria such as Eisenstein criterion and EisensteinDumas criterion. Another criterion in the literatures is known as Newton polygon method. Recently, Gao has strengthened Newton polygon method as Newton polytope method. As a result of Newton polytope method, existence of an integrally indecomposable polytope in Rn implies the existence of an infinite family of absolutely irreducible polynomials in n variables over an arbitrary field (see Remark below). In [2, 3], infinite classes of integrally indecomposable polytopes have been found by some methods for constructing integrally indecomposable polytopes. A connection of homothetic indecomposability to integral indecomposability was also given in [3]. In this study we obtain further connections and applications of homothetic indecomposability to integral indecomposability. Using modifications of some methods for construction of homothetically indecomposable polytopes, we obtain new methods for construction of new families of integrally indecomposable polytopes, which are not included in [2, 3]. .
